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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Bearing defects are one of the most important mechanical sources for vibration and noise generation in machine tool spindles. In this study, an integrated finite element (FE) model is proposed to predict the vibration responses of a spindle bearing system with localized bearing defects and then the sensor placement for better detection of bearing faults is optimized. A nonlinear bearing model is developed based on Jones' bearing theory, while the drawbar, shaft and housing are modeled as Timoshenko's beam. The bearing model is then integrated into the FE model of drawbar/shaft/housing by assembling equations of motion. The Newmark time integration method is used to solve the vibration responses numerically. The FE model of the spindle-bearing system was verified by conducting dynamic tests. Then, the localized bearing defects were modeled and vibration responses generated by the outer ring defect were simulated as an illustration. The optimization scheme of the sensor placement was carried out on the test spindle. The results proved that, the optimal sensor placement depends on the vibration modes under different boundary conditions and the transfer path between the excitation and the response.

Due to the increasing demands from aerospace, automotive, die/mold and other industries, machine tools with higher speed, precision and reliability are required urgently. In a machine tool, the spindle directly affects the cutting ability of the whole machine tool, since it either carries cutting tools as in milling operations, or work-pieces as in turning. In general, there are four types of spindle depending on the type of drives used: belt drive, gear drive, direct drive and integrated (built-in) drive [

Bearings play an important role in machine tool spindle systems. Compared with hydrostatic, aerostatic or magnetic bearings [

Most studies have focused on feature extraction of the faulty bearing from vibration signals, however, just a few of works attempted to model bearing faults mathematically so that the vibration signals generated by the defects can be simulated and explained theoretically. McFadden and Smith [

In the above research, rolling element bearings are modeled linearly and usually simplified as single DOF systems. However, due to the nonlinear Hertzian force/deformation relationship, the varying compliance vibration effect, and lubricant film effect, rolling ball bearings show high non-linearity and time-varying characteristics during operation, so a linear model is not adequate to express the dynamic behavior of bearings. Recently, Sopanen and Mikkola [

Bearings in rotating machines are mechanically coupled to supporting structures, and thus defect-induced transient signals are often masked by interfering signals or background noise [

In the machine tool spindle system, bearings work as a subsystem. Accurately predicting the vibration signals of the spindle with bearing defects remains a challenging task because of its complicated nonlinear behavior. In this paper, the machine tool spindle system is focused on and an integrated finite element (FE) model for a spindle-bearing system is proposed. The nonlinear bearing model is developed based on Jones' bearing theory [

The rest of the paper is organized as follows: in Section 2, a dynamic model of a spindle-bearing system is given with experimental validation. Localized bearing faults modeling is presented in Section 3, followed by the optimization of sensor positions in Section 4. The conclusions are given in Section 5.

The machine tool spindle system usually consists of drawbar, shaft, housing, bearings and other accessories. A typical test spindle is shown in

The spindle drawbar/shaft/housing and the angular contact ball bearing are modeled separately. The drawbar, shaft, and housing are modeled as Timoshenko's beam, while the nonlinear bearing model is developed based on Jones' bearing theory which considers the centrifugal force and gyroscopic effects from rolling balls. The bearing model is integrated into the FE model of shaft/housing by assembling equations of motion. The pulley, clamping unit and other accessories are modeled using rigid disk elements. The spacers between bearings are modeled using bar elements. This FE modeling process can be extended to high-speed spindles of integrated structure, in which the motor is modeled using rigid disk elements. With the FE model of the spindle, the Newmark integration method is used to obtain the vibration responses numerically.

The Timoshenko beam element (as shown in _{x}, δ_{y}, δ_{z}_{y}, γ_{z}

The equation of motion for the drawbar/shaft in matrix form is expressed by including the centrifugal force and gyroscopic effects as:
_{b}_{b}_{b}_{C}_{b}

In the spindle system, disk and other accessories (e.g., clamping units, nuts) are commonly used, and they are modeled using rigid disk elements. The equation of motion is given as:
_{d}_{d}_{d}

The geometric drawing of an angular contact ball bearing is shown in _{i}_{o}_{th} ball is:

Similar to the shaft/housing, the bearing is modeled with ‘bearing element’. Each ‘bearing element’ consists of two nodes—the inner ring node and the outer ring node. The motion vectors of bearing nodes are expressed by _{i}_{o}

Under the working condition, the bearing rotates at high speed with applied axial loads and/or radial loads and the relative positions between the inner ring, ball and outer ring change consequently. Based on Jones' bearing model, the geometry relations of the bearings in the X-Y plane are shown in

From _{ik}_{ik}_{ik}_{ik}, δ_{ik}_{ok}

Considering the plane passing through the bearing axis and the center of a ball located at azimuth _{k}

From

Combining _{1}, _{2}, _{3} and _{4} are error variables. The Newton-Raphson iteration method is used to solve the _{bk}, Y_{bk}, δ_{ik}, δ_{ok}_{i}_{o}

The forces. _{i}_{xi}, F_{xi}, F_{zi}, M_{yi}, M_{zi}^{T} applied on the inner ring of the bearing are given as:

Similarly, the forces _{o}_{xo}, F_{xo}, F_{zo}, M_{yo}, M_{zo}^{T} applied on the outer ring of the bearing can be obtained. The bearing stiffness matrix is obtained by calculating the derivative of the force _{x}, F_{x}, F_{z}, M_{y}, M_{z}^{T} acting on the bearing rings with respect to the displacement _{x}, δ_{y}, δ_{z}, γ_{y}, γ_{z}^{T} of bearing rings, namely:

The bearing stiffness matrix _{B} has the form:
_{xx}_{yy}_{zz}_{θxθx}_{θyθy}

The finite element model of the spindle-bearing system is shown in

By assembling the equations of the drawbar/shaft/housing and bearings, the following general nonlinear dynamic equation for the spindle-bearing system is obtained:
_{b}, _{s}_{b}_{b}_{B} − Ω^{2}_{C}_{s}_{B} depends on the displacement,

If the external load is time-varying, then the system responses are time-varying as well and the equation of motion at time

Assuming that the displacement, velocity, and acceleration vectors at time _{t+Δt}, velocity _{t+Δt} and acceleration _{t+Δt}) at time

The test spindle was hung using elastic strings as a free-free system. The impact forces were applied by a hammer on the spindle nose in the radial direction and the vibration responses at the opposite part of the nose were recorded by an accelerometer, as shown in

In the simulation, some parameters must be identified. These parameters include joint stiffness between the drawbar and the shaft, and the system damping ratios. It is possible to obtain a good match between the simulation and the measurement by tuning these parameters. The joint stiffness was found to be 2 × 10^{8} N/m in the radial direction. The modal damping ratios were borrowed from experimental data and used in the FE simulations. From the simulated FRFs, the first mode (

The simulated and experimentally measured FRFs at the spindle nose are shown in

The time-history response of the acceleration

The acceleration time responses and their spectra at the front side and the rear side of the spindle housing are predicted in

If bearing defects occur in the spindle system, the defect in one surface of a bearing strikes another surface and then an impact force is generated, which may excite resonances in the spindle-bearing system. With the input excitation from an appropriate bearing fault model, the simulation of vibration response caused by the bearing defects is feasible.

Vibration signals measured on a spindle contain rich physical information about the operating conditions. When local defects (e.g., cracks, pits, spalls) exist in one of the bearing components, transient impact forces occur whenever such a defect on one surface strikes its mating surface. The impact forces will excite the vibration responses at the natural frequencies of bearing parts and housing structure. During the rotating process, a series of approximately equally spaced force impulses are produced. The repetition rate of the force impulses is equal to the characteristic frequencies of the bearing,

Similar to the impact force generated by a hammer test, the shape of the force impulse is modeled as a triangular form, as shown in _{r}_{r}

To explain the localized bearing faults modeling process, the outer ring defect of the first bearing (HYKH61914) in the spindle is modeled as an example. The bearing parameters are listed in

The load is applied in the radial direction and there is a pit on the surface of the outer ring, as shown in

The successive force impulses generated by the localized defect on the outer ring is shown in

The impulse responses due to the outer ring defect are displayed in

The characteristic frequencies of the outer ring defect are shown in

For the spindle-bearing system, out-of-balance contributions at the shaft rotational frequency _{r}, is always evident as various sources (e.g., shaft mis-alignment) contribute to unbalanced rotation of the shaft. To include the out-of-balance effect, the vibration responses induced by force impulses are modulated in amplitude, as shown in

The characteristic frequencies of the outer ring defect with the out-of-balance effect are shown in _{r} (20 Hz) is extracted as well in

For measurements in practice, the noise problem can never be avoided. For enhancing the simulation towards more realistic data, the impulse responses are contaminated with a normal distribution

The characteristic frequencies of the outer ring defect with the out-of-balance and the noise contamination effects are shown in

For the condition monitoring and diagnosis of the spindle bearings, vibrations produced due to defects are usually measured using the velocity transducer or accelerometers. However, the measured vibration responses are affected by the transmitting media between defects and sensors. To obtain a better monitoring effect, the optimization of the sensor position is necessary.

In this study, the optimization scheme of the sensor position is carried out on the test spindle, as shown in

It is well known that the displacement of a vibratory system due to the excitation is related with mode shapes. In

On the other hand, it is better to monitor the vibration response signals at the nearest location to the defective bearing from experience [

The acceleration responses due to the localized defect are simulated by using the dynamic spindle-bearing model. The dynamic responses at different positions are simulated and then compared with each other to obtain an optimal sensor position. The peak-to-peak value of the vibration responses at each measured point is displayed in

Although the point 4 is the nearest point to the defective bearing, it is also close to the fixed flange which lowers the vibration response. For point 17, although the displacement of the mode shapes is the largest, the vibration energy decreases largely due to the long transfer path between the excitation and the response. It can be concluded that, the optimal sensor placement depends on the vibration modes under different boundary conditions, and the transfer path between the excitation and the response.

The impulse responses of the outer ring defect at three typical measured points (1, 4 and 17) are shown in

Next, the noise contamination is included to enhance the simulation towards more realistic data. The impulse responses with noise contamination at the three measured points (1, 4 and 17) are displayed in

The envelope spectra of the impulse responses with noise are shown in _{r} (20 Hz) is extracted as well (

In the current study, the vibration response simulation and sensor placement optimization of machine tool spindles were investigated using an integrated FE model. The localized bearing defects in spindles were modeled visually, and vibration responses generated due to the outer ring defect were simulated as an illustration. The results have shown that:

The FE model of the spindle is capable of predicting the acceleration time responses due to the excitation. With the input excitation from an appropriate bearing fault model, the simulation of vibration response caused by bearing defects in machine tool spindles is feasible.

The noise decreases the amplitudes at the bearing characteristic frequencies in the envelop spectrum. If the signal-to-noise ratio is very low, the characteristic frequencies are submerged and cannot be detected from sensor signals.

The optimization of sensor placement in machine tool spindles depends on the vibration modes under different boundary conditions and the transfer path between the excitation and the response.

The authors wish to express their heartfelt gratitude to Prof. Altintas from Manufacturing Automation Laboratory (MAL), The University of British Columbia, for his valuable instruction, guidance, and support throughout this work. The experiments in this paper were carried out in MAL. The authors also thank Yuzhong Cao for his guidance and generous help. This work is jointly supported by National Natural Science Foundation of China (Grant No. 51105294, 51035007), Research Fund for the Doctoral Program of Higher Education of China (20110201120029) and the Fundamental Research Funds for the Central University.

The test spindle.

Timoshenko beam element.

Geometry of an angular contact ball bearing.

Positions of ball center and raceway groove curvature centers.

The force acting on the ball at angular position _{k}

The finite element model of the spindle-bearing system.

The hammer test of the spindle. (

Simulated and experimentally measured FRFs at the spindle nose in the free-free condition.

The measured impact force at the spindle nose.

The acceleration time responses and spectra at the front and the rear side of the spindle housing: (

The force impulse form.

The localized defect on the outer ring.

The successive force impulses.

The impulse responses of the outer ring defect.

The characteristic frequencies of the outer ring defect. (

The amplitude modulated impulse responses of the outer ring defect.

The characteristic frequencies of the outer ring defect with the out-of-balance effect. (

The amplitude modulated impulse responses of the outer ring defect with noise contamination.

The characteristic frequencies of the outer ring defect with the out-of-balance and the noise contamination effects. (

The sensor distribution on the test spindle.

The first three mode shapes of the spindle (The flange is fixed).

The peak-to-peak value of vibration responses.

The noise-free impulse responses of the outer ring defect. (

The impulse responses of the outer ring defect with noise. (

The envelop spectrum of the impulse responses with noise. (

Parameters of the fault bearing.

HYKH61914 | 85 | 6.35 | 32 | 25 |